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REFLECTION SUBGROUPS of COXETER GROUPS 1. Introduction In TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 2, February 2010, Pages 847–858 S 0002-9947(09)04859-4 Article electronically published on September 18, 2009 REFLECTION SUBGROUPS OF COXETER GROUPS ANNA FELIKSON AND PAVEL TUMARKIN Abstract. We use the geometry of the Davis complex of a Coxeter group to investigate finite index reflection subgroups of Coxeter groups. The main result is the following: if G is an infinite indecomposable Coxeter group and H ⊂ G is a finite index reflection subgroup, then the rank of H is not less than the rank of G. This generalizes earlier results of the authors (2004). We also describe the relationship between the nerves of the group and the subgroup in the case of equal rank. 1. Introduction In [3], M. Davis constructed for any Coxeter system (G, S) a contractible piece- wise Euclidean complex, on which G acts properly and cocompactly by reflections. In this paper, we use this complex to study finite index reflection subgroups of in- finite indecomposable Coxeter groups from a geometrical point of view. We define convex polytopes in the complex to prove the following result: Theorem 1.1. Let (G, S) be a Coxeter system, where G is infinite and indecom- posable, and S is finite. If P is a compact polytope in Σ, then the number of facets of P is not less than |S|. This generalizes results of [7], where a similar result was proved for fundamental polytopes of finite index subgroups of cocompact (or finite covolume) groups gener- ated by reflections in hyperbolic and Euclidean spaces. M. Dyer [5] proved that any reflection subgroup of a Coxeter group is also a Coxeter group. Using Theorem 1.1 and a criterion for finiteness of a Coxeter group provided by V. Deodhar [4], we obtain the main result of this paper: Theorem 1.2. Let (G, S) be a Coxeter system, where G is infinite and indecom- posable and S is finite. Let H ⊂ G be a finite index reflection subgroup. Then any set of reflections generating H contains at least |S| elements. Further, we consider the geometry of the Davis complex itself. E. M. Andreev [1] obtained the following result for polytopes in hyperbolic and Euclidean spaces. Received by the editors January 15, 2008. 2000 Mathematics Subject Classification. Primary 20F55, 51M20; Secondary 51F15. The first author was supported in part by grants NSh-5666.2006.1, INTAS YSF-06-10000014- 5916, and RFBR 07-01-00390-a. The second author was supported in part by grants NSh-5666.2006.1, MK-6290.2006.1, INTAS YSF-06-10000014-5916, and RFBR 07-01-00390-a. c 2009 American Mathematical Society 847 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 848 ANNA FELIKSON AND PAVEL TUMARKIN Theorem 1.3 (Andreev [1]). Let P be an acute-angled polytope in En or Hn,and let a and b be two faces of P .Ifa ∩ b = ∅, then the minimal planes containing, respectively, a and b are also disjoint. In Section 4, we define acute-angled polytopes in the Davis complex and prove a counterpart of Andreev’s theorem for non-intersecting facets of acute-angled poly- topes (see Lemma 4.3). InSection5,wefocusonthecaseofaCoxetergroupG having a finite index reflection subgroup H of equal rank; i.e., this is the minimum possible rank for such an H. We show that the nerve of H can be obtained from the nerve of G by deleting some simplices (Lemma 5.1). We also give conditions on the combinatorics of (G, S) which are necessary in order for G to contain such an H (Lemma 5.4). We would like to thank E. B. Vinberg for useful comments and discussions and R. B. Howlett for communicating a proof of Lemma 2.1. The work was mainly done during our stay at the University of Fribourg, Switzerland. We are grateful to the University for hospitality. 2. Preliminaries 2.1. Coxeter systems. A Coxeter group is a group with presentation mij S | (sisj ) =1 for all si,sj ∈ S,wheremii =1andmij ∈{2, 3,...,∞} for i = j.Wefurther require S to be finite. S is called a standard generating set. A pair (G, S) is called a Coxeter system. If S is fixed, G is called decomposable if S = S1 ∪ S2,whereS1 and S2 are non-empty subsets such that sisj = sjsi for all si ∈ S1, sj ∈ S2.IfG is not decomposable, it is called indecomposable. An element s ∈ G is called a reflection if it is conjugate to some of the si ∈ S (in particular, any si ∈ S is a reflection). A subgroup H ⊂ G is called a reflection subgroup of G if H is generated by reflections. For any T ⊂ S a reflection subgroup GT generated by all si ∈ T is called a standard subgroup of G. The rank of a Coxeter system (G, S) is the number of reflections in S.Wedenote it by |S|. The proof of the following lemma is suggested by R. B. Howlett. Lemma 2.1. Let (G, S) be a Coxeter system of rank n.ThenG cannot be generated by less than n reflections. Proof. Let S = {s1,...,sn}. Consider the Tits representation of (G, S)onareal vector n-space V (see e.g. [2]). Suppose that G is generated by k<nreflections r1,r2,...,rk along vectors v1,v2,...,vk, i.e. (vi,x) ri(x)=x − 2 vi. (vi,vi) Let L be the linear subspace spanned by v1,...,vk.Letg = ri1 ri2 ...ril be any element of G. We prove by induction on l that for any v ∈ V , g(v) ∈ v + L.For l = 0 the statement is evident (since v ∈ v + L). Suppose the statement holds for License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REFLECTION SUBGROUPS OF COXETER GROUPS 849 all elements g = ri1 ri2 ...ril−1 .Letg = ri1 g1,whereg1 = ri2 ...ril .Then (v ,g (v)) − i1 1 ∈ ⊂ g(v)=ri1 g1(v)=g1(v) 2 vi1 v + L + λvi1 v + L (vi1 ,vi1 ) (where λ ∈ R). Hence, g(v) ∈ v + L for any g ∈ G. Suppose that g ∈ G is a reflection along some vector v.Then−v = g(v) ∈ v +L. Therefore, v ∈ L. Hence, by the construction of the Tits representation of (G, S), L should coincide with V , which is impossible for a space spanned by k<n vectors. 2.2. Davis complex. For any Coxeter system (G, S) there exists a contractible piecewise Euclidean cell complex Σ (G, S) (called the Davis complex)onwhichG acts discretely, properly and cocompactly. The construction was introduced by Davis [3]. In [9] Moussong proved that this complex yields a natural complete piecewise Euclidean metric which is CAT (0). We give a brief description of this complex following [10]. For a finite group G the complex Σ (G, S) is just one cell, which is obtained as a convex hull C of G-orbit of a suitable point p in the standard linear representation of G as a group generated by reflections. The point p is chosen in such a way that its stabilizer in G is trivial and all the edges of C are of length 1. The faces of C are naturally identified with Davis complexes of the subgroups of G conjugate to standard subgroups. If G is infinite, the complex Σ (G, S) is built up of the Davis complexes of maxi- mal finite subgroups of G gluing together along their faces corresponding to common finite subgroups. The 1-skeleton of Σ (G, S) considered as a combinatorial graph is isomorphic to the Cayley graph of G with respect to the generating set S. The action of G on Σ (G, S) is generated by reflections. The walls in Σ (G, S) are the fixed point sets of reflections in G. The intersection of a wall α with cells of Σ (G, S) supplies α with a structure of a piecewise Euclidean cell complex with finitely many isometry types of cells. Walls are totally geodesic: any geodesic joining two points of α lies entirely in α. Since Σ is CAT (0), any two points of Σ may be joined by a unique geodesic. Any wall divides Σ (G, S) into two connected components. The complement of the union of all walls in Σ (G, S) is a disjoint collection of connected components, and the closure of each of these components is compact; these closures are called chambers. Any chamber is a fundamental domain of G-action on Σ (G, S). The set of all chambers with appropriate adjacency relation is isomorphic to the Cayley graph of G with respect to S. A nerve of a Coxeter system (G, S) is a simplicial complex with vertex set S. A collection of vertices spans a simplex if and only if the corresponding reflections generate a finite group. It is easy to see that the combinatorics of the chamber of Σ(G, S) is completely determined by the nerve: a collection of walls of a chamber has a non-empty intersection if and only if the corresponding reflections generate a finite reflection group. For any subgroup H ⊂ G we say that a wall α is a mirror of H if H contains the reflection with respect to α. In what follows, if G and S are fixed, we write Σ instead of Σ (G, S). 2.3. Convex polytopes in Σ. For any wall α of Σ we denote by α+ and α− the closures of the connected components of Σ\α; we call these components halfspaces.
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